But it was OK in the Old Testament, wasn’t it?

David, and all other men in the Old Testament up to and including the Jews in Jesus’ day who either divorced their wives and took another or added another wife to their already intact relationship would today be guilty of adulterous polygamy under the Atonement. 

This was not the case under the Old Covenant, however. Jesus tells the religious leaders that Moses wrote this law allowing divorce because of the hardness of people's hearts.  Who did Jesus say wrote that law? God? No, Moses. Jesus says it was not this way from the beginning. In other words, the separation of spouses joined by God is not God's will for married couples, period. 

He says that Moses wrote the law, the inference is with the allowance of God. Why allow such a thing? It seems reasonable to interpret His explanation to mean this was allowed to be written because He knew that people were, to put it plainly, stubborn, sinful jerks and since they lived among heathen lands who practiced this and they were eventually going to do this anyway it is best if it is regulated to protect from even further abuse the women and children that result. 

However, just like He did with laws about incest and close-family polygamy being added to the OT roster LATER meaning those who came before would not be judged guilty (such as Cain marrying one of his sisters, a daughter of Adam and Eve, or Jacob marrying two sisters at the same time, both of which are direct violations of later laws given by God), Jesus being the fulfillment of all the Law and Elohim, the Creator of the Universe and the Law-Giver, Jesus alone is qualified to add to or take away from His own words. In the same manner that He clarifies and even heightens the Law surrounding adultery, revenge-taking and murder elsewhere in the Gospel accounts, He irrevocably makes the God-head's position clear about divorce. 

Some proponents of Christian polygamy have said that these verses show that marrying another spouse without divorcing the first spouse is not adultery and it only becomes adultery if the first spouse is divorced. This does not stand up to Jesus' other assertions about lustful looks towards women being adultery but that aside it also can only work if one deliberately forgets the topic Jesus is actually dealing with in this discourse - divorce.  Jesus, as God, is saying that God does not recognize divorce as a means of breaking a marriage covenant Hs has put together and therefore He does allow/join a second, new marriage covenant to become a valid spiritual aka "one flesh" union.

So, Jesus says that regardless of the addition of man's law to God's law through Moses in Old Testament times, the Creator incarnate says that in God's eyes there is no annulment of a marriage union even if a Jewish man obtains a civilly legal divorce paper which they called a "get".  If there ever truly was before, after Jesus there is now no such thing as spiritual divorce.  In much the same manner that previously lawful human relations such as incest and the eating of various animals became unlawful in the fullness of time when Elohim changed His laws to reflect His will, Yeshua states that divorce is not something that Yahweh acknowledges in the spirit.  From that moment on, the utterance became the higher Law and Jesus thus clarifies with his disciples later that the man who gets a legal Get for his first wife to dissolve the union and then goes on to legally marry a second wife commits adultery against his first wife and likewise that a first wife who is told to "get away" and she then marries a second husband is actually caused to become an adulteress with this legally married second husband in God's eyes, although in the eyes of the world both are considered legally innocent and under the Old Covenant they would also have been considered morally innocent even if she were part of a harem or plural wives! 

Therefore the question of God's position on polyamorous unions after the time of Christ becomes clear - Marrying a second spouse while the first spouse is still alive is adultery. Whether divorced or not it is irrelevant because Jesus says, essentially, there's no such thing as divorce. In reality then, Jesus is literally talking about polygny and polyandry - a married man marries a second wife, his first wife is alive therefore he is doing this while still married to the first wife = adultery; a married woman legally marries a second husband, she is doing this while her first husband is still alive therefore she is doing this while still married to the first husband = adultery.  God says that divorcees are actually living in polyamorous marital affairs and that in His eyes this breaks the seventh commandment and causes the already married one to commit adultery either against the first (real) wife by marrying another woman or against the first (real) husband by becoming an adulterous woman by marrying another man.

Further Jesus says that even though Moses in the Law allowed divorce in the past Jesus considers remarriage after divorce adultery (breaking one of the Ten). Following that kind of new revelation that Jesus shared then other relationship situations which existed in the OT and were even regulated in the Law and not rebuked by God (like plural marriage) would fall into the same category. Jesus set up a higher law than Moses in the respect of marriage.